Let R1 = {(a, b) ∈ N × N : |a – b| ≤ 13} and R2 = {(a, b) ∈ N × N : |a – b| ≠ 13}. Then on N:
Both R1 and R2 are equivalence relations
Neither R1 nor R2 is an equivalence relation
R1 is an equivalence relation but R2 is not
R2 is an equivalence relation but R1 is not
The Correct Option is B
Solution and Explanation
The correct answer is (B) : Neither R1 nor R2 is an equivalence relation
R1 = {(a, b) ∈ N × N : |a – b| ≤ 13} and
R2 = {(a, b) ∈ N × N : |a – b| ≠ 13}
In R1: ∵ |2 – 11| = 9 ≤ 13
∴ (2, 11) ∈ R1 and (11, 19) ∈ R1 but
(2, 19) ∉ R1
∴ R1 is not transitive
Hence R1 is not equivalence
In R2 : (13, 3) ∈ R2 and (3, 26) ∈ R2 but
(13, 26) ∉ R2 (∵ |13 – 26| = 13)
∴ R2 is not transitive
Hence R2 is not equivalence.
Top Questions on sets
- If \( a>0, b>0, c>0 \) and \( a, b, c \) are distinct, then \( (a + b)(b + c)(c + a) \) is greater than:
- The modulus of the complex number \( z \) such that \( |z + 3 - i| = 1 \) and \( \arg(z) = \pi \) is equal to:
- In a statistical investigation of 1003 families of Calcutta, it was found that 63 families have neither a radio nor a TV, 794 families have a radio, and 187 have a TV. The number of families having both a radio and a TV is:
- Number of subsets of set of letters of word 'MONOTONE' is:
- Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \). If \( f : X \rightarrow Y \) given by \( f(x) = -8(x + 5)^2 \) is one-to-one, then the codomain \( Y \) is:
Questions Asked in JEE Main exam
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
Read More: Set Theory
Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
Read Also: Set Operation
Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
Read More: Types of Sets